Some excellent advice here from the British Association for Shooting and Conservation (BASC) on hunting rabbits with air rifles:
The BASC says that rabbits must not be shot less than two metres from their burrows. This is so as to prevent the possibility of a wounded rabbit getting back to its burrow before you can dispatch it. This is very good advice.
This evening I stalked up to a warren where I could see some rabbits. On my approach they went back to earth but I lay flat and waited for ten minutes. One emerged slowly and sat just above the entrance to its burrow and in plain sight for an easy 30-yard shot. But a shot rabbit on its burrow will very likely- even if it's killed outright - just fall straight back down the hole and be lost. This rabbit paused and then ran straight across the field to enter another burrow on the other side. I carried on waiting and five minutes later another rabbit emerged. Again, it waited, then it hopped a couple of metres away from its hole - and then sat up and offered me its profile. I had plenty of time to sight and stabilise; I took the head shot and it went down instantly and lay still. I got up to go and retrieve the rabbit - utterly confident that the shot was a good one -but then saw to my distress that it was not in fact dead - it had started to struggle back towards its burrow and it made it before I could get there or take another shot.
I can certainly testify that this is a very bad feeling: knowing that the animal is probably seriously wounded and also that there's nothing you can do about it. I walked back home across the fields sick at heart and angry with myself for having perhaps grabbed too greedily at the shot.